## 35.20 Germs of schemes

Definition 35.20.1. Germs of schemes.

A pair $(X, x)$ consisting of a scheme $X$ and a point $x \in X$ is called the *germ of $X$ at $x$*.

A *morphism of germs* $f : (X, x) \to (S, s)$ is an equivalence class of morphisms of schemes $f : U \to S$ with $f(x) = s$ where $U \subset X$ is an open neighbourhood of $x$. Two such $f$, $f'$ are said to be equivalent if and only if $f$ and $f'$ agree in some open neighbourhood of $x$.

We define the *composition of morphisms of germs* by composing representatives (this is well defined).

Before we continue we need one more definition.

Definition 35.20.2. Let $f : (X, x) \to (S, s)$ be a morphism of germs. We say $f$ is *étale* (resp. *smooth*) if there exists a representative $f : U \to S$ of $f$ which is an étale morphism (resp. a smooth morphism) of schemes.

## Comments (0)